A098081 a[n]=1+1/Log[Gamma[2-a[n-1]]]:

0, 7, 1, 18, 0, 55, 1, 295, 0, 2412, 1, 28397, 0, 455264, 1, 9487328, 0, 247621547

Offset

0,2

Comments

My idea is to invent a fractional Lyapunov exponent: L(s)=Sum[ Log[Abs[ d^sx[n]/dt^s]],{n,1,m}]/m where: Gamma[2]=1 d^sx[n]/dt^s=Gamma[2]*x[n]^(1-s)/Gamma[2-s] Or L(s)=Sum[ Log[Abs[x[n]^(1-s)/Gamma[2-s]],{n,1,m}]/m If the average of x[n] is one then or Log[1]=0: L[s]=Log[Gamma[2-s]] which is always negative since: 0<=Gamma[2-s]<=1 on 0<=s<=1 which gives a Kaplan -Yorke dimension of: (d0 the topological dimension) if the other exponents are one dky(s)=d0+1/Log[Gamma[2-s]] Which gives a roughness gap in the range: s'=1+1/Log[Gamma[2-s]]

References

Kiran M. Kolwankar and Anil D. Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6 (1996), 505-513.

Links

Table of n, a(n) for n=0..17.

Mathematica

Clear[s, n, digits] s[n_]:=1+1/Log[Gamma[2-s[n-1]]]; s[1]=0.6 digits=18 a=Table[Floor[Abs[s[n]]], {n, 1, digits}] ListPlot[a, PlotJoined->True, PlotRange->All, Axes-> False]

Crossrefs

Sequence in context: A348983 A013614 A179530 * A125234 A028325 A245484

Adjacent sequences: A098078 A098079 A098080 * A098082 A098083 A098084

Keyword

nonn

Author

Roger L. Bagula, Sep 13 2004

Status

approved